An isoptic is the set of points for which two tangents of a given curve meet at a fixed angle (see below).
An isoptic of two plane curves is the set of points for which two tangents meet at a fixed angle.
Thales' theorem on a chordPQ can be considered as the orthoptic of two circles which are degenerated to the two points P and Q.
Orthoptic of a parabolaedit
Any parabola can be transformed by a rigid motion (angles are not changed) into a parabola with equation . The slope at a point of the parabola is . Replacing x gives the parametric representation of the parabola with the tangent slope as parameter: The tangent has the equation with the still unknown n, which can be determined by inserting the coordinates of the parabola point. One gets
If a tangent contains the point (x0, y0), off the parabola, then the equation
holds, which has two solutions m1 and m2 corresponding to the two tangents passing (x0, y0). The free term of a reduced quadratic equation is always the product of its solutions. Hence, if the tangents meet at (x0, y0) orthogonally, the following equations hold:
If a tangent contains the point , off the ellipse, then the equation
holds. Eliminating the square root leads to
which has two solutions corresponding to the two tangents passing through . The constant term of a monic quadratic equation is always the product of its solutions. Hence, if the tangents meet at orthogonally, the following equations hold:
The last equation is equivalent to
From (1) and (2) one gets:
The intersection points of orthogonal tangents are points of the circle .
Hyperbolaedit
The ellipse case can be adopted nearly exactly to the hyperbola case. The only changes to be made are to replace with and to restrict m to |m| > b/a. Therefore:
The intersection points of orthogonal tangents are points of the circle , where a > b.
Orthoptic of an astroidedit
An astroid can be described by the parametric representation
From the condition
one recognizes the distance α in parameter space at which an orthogonal tangent to ċ(t) appears. It turns out that the distance is independent of parameter t, namely α = ± π/2. The equations of the (orthogonal) tangents at the points c(t) and c(t + π/2) are respectively:
Their common point has coordinates:
This is simultaneously a parametric representation of the orthoptic.
Elimination of the parameter t yields the implicit representation
Introducing the new parameter φ = t − 5π/4 one gets
Lawrence, J. Dennis (1972). A catalog of special plane curves. Dover Publications. pp. 58–59. ISBN 0-486-60288-5.
Odehnal, Boris (2010). "Equioptic Curves of Conic Sections" (PDF). Journal for Geometry and Graphics. 14 (1): 29–43.
Schaal, Hermann (1977). Lineare Algebra und Analytische Geometrie. Vol. III. Vieweg. p. 220. ISBN 3-528-03058-5.
Steiner, Jacob (1867). Vorlesungen über synthetische Geometrie. Leipzig: B. G. Teubner. Part 2, p. 186.
Ternullo, Maurizio (2009). "Two new sets of ellipse related concyclic points". Journal of Geometry. 94 (1–2): 159–173. doi:10.1007/s00022-009-0005-7. S2CID 120011519.